Reposted Article
Recalling Yin and Yang
Original Text
This article was trans-coded by SimpRead; original source: www.zhihu.com
Mundane烟火 (Human World’s Vitality)
What’s truly strange about this question isn’t multiplication.
It’s the word “almost.”
Why do almost all theories—after circling around in complex ways—ultimately take on the same form?
Newton’s did.
Maxwell’s did.
Einstein’s did.
Quantum mechanics did too.
You’d think they discovered entirely different worlds.
Yet when written down, their equations look as if penned by the same person across different eras.
At first, I assumed this was simply because natural laws inherently assume that form.
Later, I realized that answer was far too lazy.
Then, I even began suspecting: maybe nature doesn’t “prefer” multiplication.
Maybe anything capable of becoming a “law” inevitably ends up taking multiplicative form.
Note: not just physics—any law.
This idea came from an unexpected place.
One day, it suddenly struck me: addition is actually quite picky.
It only adds things of the same kind.
Apples can only be added to apples.
Lengths only to lengths.
Energy only to energy.
If you try adding mass and velocity, you immediately know it’s wrong.
Why?
I used to think: because their dimensions differ.
Later, I realized dimensionality is merely the consequence, not the cause.
The real reason is that they belong to two entirely different worlds:
One describes what something is.
The other describes how it changes.
They coexist compatibly—but cannot substitute for each other.
That’s when it dawned on me: addition has always been doing just one thing.
Acknowledging equivalence.
Only things fully interchangeable can be added.
Multiplication, by contrast, operates precisely between things that cannot substitute for each other.
Mass cannot become velocity.
Space cannot become time.
Charge cannot become electric field.
Precisely because they differ, they can multiply.
At that moment, I suddenly saw that addition and multiplication aren’t just two arithmetic operations.
They’re two philosophies.
Addition says:
“You belong to the same category.”
Multiplication says:
“You don’t.”
Physics instantly felt profoundly unified.
All addition occurs within compatible domains.
All multiplication occurs where domains meet.
Then another stranger question occurred to me:
Why only these two?
Why not a third?
Why shouldn’t there exist some operation—neither addition nor multiplication—that serves as physics’ foundation?
I used to reply: “Because mathematics works that way.”
Later, I realized that statement says nothing at all.
What truly changed my mind was category theory.
Category theory offers an unusually austere perspective.
It barely cares what objects are.
It cares only whether objects remain distinguishable from one another.
As long as distinguishability persists, reality can be decomposed.
And decomposition inevitably yields exactly two phenomena:
One called juxtaposition.
The other called pairing.
Only later did I realize: these are addition and multiplication.
Nearly all mathematics falls under these two operations.
Objects can lie side-by-side.
Or they can bind together.
Vector spaces do so.
Groups do so.
Rings do so.
Topology does so.
Logic does so.
Even programming does so.
Then something chilling hit me:
We’ve always assumed mathematics invented addition and multiplication.
But perhaps not.
Perhaps any world capable of expressing “divisibility” spontaneously generates these two operations—not because mathematics dictates them, but because division and connection exhaust all logical possibilities.
At least within rational frameworks.
You divide the world—this is addition.
You re-establish relationships among divided parts—this is multiplication.
Where is the third?
There isn’t one.
Because beyond these two, what else can you possibly do?
Thus I began to understand why many mathematicians increasingly ignore numbers.
They care about structure.
Numbers are merely one possible realization.
Addition and multiplication are only shadows cast by structure.
What truly exists are two universal relations:
One preserving independence.
The other creating connection.
At that point, logarithms’ uncanny power suddenly made perfect sense.
As children, teachers told us logarithms invert exponentiation.
Later, learning more, I sensed that wasn’t quite right.
Their true power lies in translating one world into another.
What happens multiplicatively in one world…
Becomes purely additive in the logarithmic world.
Growth becomes accumulation.
Proportion becomes difference.
Scale becomes distance.
Conversely, exponentiation translates all addition back into multiplication.
I increasingly feel logarithms aren’t functions at all—
They’re translators.
Standing between two universes.
People on the left speak multiplication.
People on the right hear addition.
Then an even more unsettling realization struck me:
Perhaps the world contains no addition or multiplication per se.
Only the same underlying structure, manifesting at different levels.
At this level, it’s called addition.
At the next level up, it’s multiplication.
At this level, it’s multiplication.
At the level below, it becomes addition again.
Thus reality exhibits infinite recursion.
You think you’re studying operations—
but you’re really studying how different levels interpret each other.
Finally, I circled back to physics.
I grew increasingly fond of a nearly forgotten phrase:
“Compatible → therefore addable.”
“Encountering → therefore multipliable.”
This statement matters more than any formula.
Two forces don’t repel each other → thus superpose.
Two probability amplitudes inhabit the same space → thus superpose.
Two particles interact → thus couple.
Position and momentum are conjugate → thus dynamics must be expressed as pairings.
You’ll find multiplication is never for calculation.
It signals:
“A mutual definition has just occurred here.”
An object gains meaning only through another object.
Eventually, I even doubted whether we named “multiplication” poorly.
It evokes multiplication tables.
Elementary school.
Repeated addition.
But multiplication in physics shares almost no kinship with elementary-school multiplication.
Its true name should be:
“Relation.”
So the original question finally flipped:
Why are physical laws almost always multiplicative?
Because physics has never studied objects.
Objects are merely frozen snapshots left behind after relations solidify.
Physics has always studied how relations generate relations.
And any mathematics capable of expressing relation inevitably grows a “multiplication.”
Any mathematics capable of expressing independence inevitably grows an “addition.”
So the world doesn’t possess just one addition and one multiplication.
Rather, every logic has its own addition and multiplication;
every mathematics has its own compatibility and coupling;
and at each ascending level of abstraction, addition and multiplication mutually transform.
Today’s multiplication becomes tomorrow’s addition.
Today’s object becomes tomorrow’s relation.
What remains invariant is never the operation itself—
but that reality performs only two fundamental acts:
Allowing independence.
Allowing encounter.
Physics, then, is merely the search—for each level—to discover what those two acts are named.
North Sea Has Fish
To answer this question, begin with a more fundamental fact:
Mathematics isn’t decoration for physics—it’s the language of physical structure. A formula’s shape depends on the intrinsic structure of the phenomenon it describes.
Nature’s foundations, in fact, consist of only a few recurring “organizational patterns.” Each pattern corresponds to a specific mathematical tool. The various formulaic forms we observe are merely different facets of these foundational structures.
Below, I map these correspondences.
I. Nature’s Foundational Organizational Patterns
1. Independence — Corresponds to Multiplication
When two non-interfering systems coexist, their joint state count equals the product of individual state counts (W_1 \cdot W_2), and their joint probability equals the product of individual probabilities. This reflects a deep mathematical truth: independence implies multiplication.
This explains why formulas linking different kinds of physical quantities almost invariably adopt multiplicative forms—e.g., pressure, volume, and temperature (PV = nRT); force, mass, and acceleration (F = ma); voltage, current, and resistance (V = IR).
2. Conservation & Superposition — Corresponds to Addition
Energy, momentum, and charge remain constant in closed systems. “Constant sum” is inherently additive. When multiple factors contribute independently without interference, total effect sums simply. Force composition, electric field superposition, and energy decomposition all follow this principle.
Underpinning addition is the linear superposition principle—a cornerstone enabling clear description of classical physics. Once systems become strongly nonlinear (e.g., intense gravity, turbulence), superposition fails—and physics becomes intractable.
3. Locality — Corresponds to Differential Equations
Physical interactions don’t “jump across space” instantaneously; they propagate stepwise through adjacent points. This “neighborhood-only” property maps mathematically to derivatives—the derivative being the rate of infinitesimal change within an infinitely small neighborhood.
Thus Maxwell’s equations, Schrödinger’s equation, fluid dynamics equations, and Einstein’s field equations are all differential equations. They describe not instantaneous relationships, but how fields continuously evolve across space.
4. Evolution & Causality — Corresponds to Time Derivatives
“The present state determines the next state”—this causal structure mathematically manifests as first-order time-derivative equations. Hamilton’s equations and the time-evolution part of Schrödinger’s equation both adopt this form. A deterministic physical universe is fundamentally a set of time-differential equations.
5. Cumulative Effects — Corresponds to Integration
When effects accumulate along paths or over time, integration applies. Work equals force integrated over displacement; action equals the Lagrangian integrated over time. Integration is essentially the “continuous version of addition.”
6. Optimization — Corresponds to Variational Principles
Nature seems to “prefer the most economical path”: light follows the shortest optical path (Fermat’s principle); particles follow paths of minimal action (Hamilton’s principle). This global optimality structure is expressed via calculus of variations: \delta S = 0.
Remarkably, entire frameworks—classical mechanics, quantum field theory, general relativity—can all be derived from a single variational principle. This represents one of physics’ deepest unifications.
7. Self-Feedback — Corresponds to Exponential Functions
Whenever “rate of change is proportional to current quantity” (dN/dt = \lambda N), the solution is necessarily e^{\lambda t}. Radioactive decay, population growth, Boltzmann distributions—all share this structure.
This explains why e appears ubiquitously in physics—it’s the natural language of self-feedback processes.
8. Periodicity — Corresponds to Trigonometric Functions & Complex Exponentials
Systems possessing “restoring forces” (displaced from equilibrium and pulled back) yield solutions of \sin, \cos, or e^{i\omega t}. Simple harmonic oscillation, electromagnetic waves, quantum phase evolution—all follow suit. This isn’t physics’ choice—it’s the mathematical inevitability of second-order linear equations.
9. Symmetry — Corresponds to Group Theory
Isotropy of space, uniformity of time, invariance under particle exchange—these properties (“unchanged under certain transformations”) are mathematically formalized as groups.
Noether’s theorem directly links symmetry to conservation laws: time-translation symmetry → energy conservation; spatial-translation symmetry → momentum conservation; rotational symmetry → angular momentum conservation. The Standard Model of particle physics is essentially a combination of groups (U(1) \times SU(2) \times SU(3)).
10. Geometry & Curvature — Corresponds to Tensors
General relativity interprets gravity as spacetime geometry. Requiring “physical laws hold in any coordinate system” mathematically necessitates tensors. Riemannian geometry provided the ready-made language enabling Einstein to write G_{\mu\nu} = 8\pi T_{\mu\nu}.
11. Quantization & Discreteness — Corresponds to Eigenvalues of Linear Operators
Quantum systems exhibit discrete energy levels; observables are operators; measurement outcomes are eigenvalues. This “only certain values allowed” structure is mathematically the spectrum of linear operators. Hilbert spaces and matrix mechanics both originate here.
12. Randomness — Corresponds to Probability Theory
Thermodynamics and quantum mechanics contain irreducible randomness. Statistical behavior of large ensembles requires distribution functions—probability theory and measure theory provide precise language.
II. Returning to the Original Question: Why Is Multiplication Most Common?
Having surveyed the above list, we can now offer a relatively complete answer.
Multiplication appears most frequently in foundational physics formulas because it simultaneously aligns with several of the most fundamental, widespread organizational patterns:
First, dimensional closure demands it. Physical quantities carry distinct units; combining them into new quantities requires multiplication or division—the only operations altering dimensions. Addition/subtraction apply only to quantities sharing identical dimensions. Thus, whenever formulas involve different physical quantities, multiplication becomes virtually unavoidable.
Second, it corresponds to independence. Independence → multiplication is one of mathematics’ deepest facts. Many natural phenomena can be approximated as combinations of independent factors—making multiplication the most natural compositional tool.
Third, linear approximations produce it. Any smooth function near a point approximates $y \approx kx$—a multiplicative form. Many phenomena, at common scales, happen to reside in such “weak-perturbation” regimes where first-order approximation suffices—Hooke’s law, Ohm’s law, and weak-field gravity all emerge this way.
Fourth, scale symmetry enforces it. If a law requires “input scaled by n implies output scaled by n,” mathematically it must assume y = kx.
Fifth, many “multiplications” are actually disguised additions. Taking logarithms converts multiplication into addition. Entropy S = k \ln W transforms “multiplying microstates of independent systems” into “adding entropies.” Thus multiplication and addition often represent two sides of the same structural coin.
In short—multiplication’s prevalence isn’t accidental. It matches independence, scaling, and dimensional combination: nature’s most ubiquitous foundational features.
III. A Deeper Question: Why Does Mathematics Fit Physics So Perfectly?
You’ll notice the above mapping is remarkably neat: each physical organizational pattern matches precisely one mathematical structure. This echoes Eugene Wigner’s famous observation: “The unreasonable effectiveness of mathematics in the natural sciences.”
This correspondence isn’t coincidence, nor is it merely “mathematics being a universal language.” More accurately:
The physical world appears built from a few core organizational principles—symmetry, locality, conservation, independence, evolution, optimization, periodicity, feedback, geometry, discreteness, randomness—and mathematics is precisely the discipline studying such abstract structures.
Physics discovers structure in concrete natural phenomena; mathematics studies structure in abstract logical space; the two meet at the level of structure itself.
Thus, physics formulas take their particular shapes—not because physicists arbitrarily chose them, nor because mathematics imposed them—but because:
What structure nature possesses, mathematics uses corresponding forms to capture.
The formula’s shape is nature’s shape.
Core Diagram Workshop
In a 1962 lecture, Richard Feynman addressed a similar student question. His response:
We spend so much textbook time on linear, simply multiplicative physical systems for one reason alone:
“Because we can solve them.”
Even the simplest pendulum, in reality, contains a nonlinear trigonometric term \sin\theta in its force equation. But when swing amplitude is very small, physicists observe \sin\theta \approx \theta.
Thus, to obtain calculable answers, humanity substitutes simple multiplication. Once beyond this safe zone of tiny perturbations, simple multiplication fails.
This illusion is a “survivorship bias” rooted in foundational education.
Open any fluid dynamics textbook, and you’ll see real physical laws brimming with partial derivatives, integrals, and complex variable summations.
Yet another question remains: Why does simplification take multiplicative rather than additive form?
Because multiplication is the only legitimate tool for crossing “dimensional barriers.”
In 1914, physicist E. Buckingham[1] noted: you can never add 1 kg and 1 m.
To create novel concepts like “force” or “energy” in our universe, the sole legitimate method is multiplying fundamental units of different natures.
To preserve physical meaning across both sides of an equation, variables must be multiplied to construct dimensionally consistent quantities—hence physical laws’ foundational skeletons must be built from products.
Moreover, our environment mostly occupies nature’s “peace zone.”
By Taylor series definition, any complex, twisted curve—when zoomed into an infinitesimally small segment—appears linear.
Expressed mathematically, this line is a simple first-order multiplication; higher-order terms (quadratic, cubic, etc.) decay to negligible magnitudes under minute variations.
Physics rests precisely upon this approximation.
In 2021, physicist J. J. Bissell[2] proved this dynamically: he showed any physical system near stable equilibrium exhibits local energy distributions resembling a flat paraboloid.
Differentiating this paraboloid to derive force laws necessarily yields an extremely simple linear equation.
What happens when the system deviates significantly from equilibrium—when perturbations grow large?
Then simple multiplication fails. In fluid dynamics’ Navier–Stokes equations, the appearance of a term involving a variable multiplied by its own spatial derivative instantly collapses the simple world into chaotic states even supercomputers struggle to resolve.
That’s why predicting turbulence remains a million-dollar Millennium Prize Problem.
Yet even knowing these simple multiplications are merely first-order approximations, it remains astonishing that mathematics—a human-invented abstract tool for logical deduction—describes nature with such uncanny precision.
As Wigner observed in 1960: nature’s amenability to such simple mathematical language is a wondrous gift—one humanity neither fully comprehends nor deserves.
References
- ^
Buckingham, E. (1914). On physically similar systems; illustrations of the use of dimensional equations. Physical Review, 4(4), 345–376. https://doi.org/10.1103/PhysRev.4.345 - ^
Bissell, J. J. (2021). On the ubiquity of classical harmonic oscillators and a universal equation for the natural frequency of a perturbed system. American Journal of Physics, 89(12), 1094–1102. https://doi.org/10.1119/10.0005948